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Euler’s formula, a fundamental relation in mathematics, offers an opportunity to explore the properties of objects in three dimensions—and in higher dimensions as well. Along the way, we will encounter a classic of operations research: the simplex algorithm.
The field ? of complex numbers was constructed to provide solutions to every quadratic equation. Surprisingly, it also contains the solutions to all algebraic equations with coefficients in ?. In technical terms, it is algebraically closed.
The concept of a group emerged in the early 19th century to solve polynomial equations and very quickly spread to other fields, including those outside mathematics: it allows us to focus on relationships between objects rather than on the objects themselves.
The notion of a group—abstract and purely algebraic? Not at all! In geometry, it keeps us from being overwhelmed by the apparent profusion and diversity of transformations, and helps us understand the different kinds of symmetry we may encounter.
